# Numerical testing by a transfer-matrix technique of Simmons' equation for the local current density in metal-vacuum-metal junctions

## TL;DR

This study evaluates Simmons' equation for local current density in metal-vacuum-metal junctions by comparing it with transfer-matrix solutions, confirming its accuracy in certain regimes and identifying its limitations.

## Contribution

The paper provides a detailed validation of Simmons' model against numerically exact transfer-matrix solutions, correcting a key factor in the image potential energy.

## Key findings

- Simmons' equation agrees well with transfer-matrix results in practical regimes.
- The model is accurate in linear and field-emission regimes.
- It becomes invalid when the potential barrier drops below the Fermi level.

## Abstract

We test the consistency with which Simmons' model can predict the local current density obtained for flat metal-vacuum-metal junctions. The image potential energy used in Simmons' original papers had a missing factor of 1/2. Besides this technical issue, Simmons' model relies on a mean-barrier approximation for electron transmission through the potential-energy barrier between the metals. In order to test Simmons' expression for the local current density when the correct image potential energy is included, we compare the results of this expression with those provided by a transfer-matrix technique. This technique is known to provide numerically exact solutions of Schrodinger's equation for this barrier model. We also consider the current densities provided by a numerical integration of the transmission probability obtained with the WKB approximation and Simmons' mean-barrier approximation. The comparison between these different models shows that Simmons' expression for the local current density actually provides results that are in good agreement with those provided by the transfer-matrix technique, for a range of conditions of practical interest. We show that Simmons' model provides good results in the linear and field-emission regimes of current density versus voltage plots. It loses its applicability when the top of the potential-energy barrier drops below the Fermi level of the emitting metal.